


The Cost of Decreased Variance

by Bitenomnom



Series: Mathematical Proof [31]
Category: Sherlock (TV)
Genre: Alternate Universe, Friendship, Gen, John is unbelievably patient, Love, M/M, Mathematics, Sherlock deletes things he shouldn't, very unusual relationship
Language: English
Status: Completed
Published: 2012-11-01
Updated: 2012-11-01
Packaged: 2017-11-17 12:52:37
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,769
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/551771
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>Sherlock had awoken wondering whose things were in 221B; clearly, they were John Watson’s.<br/>Clearly, John Watson was his flatmate.<br/>And clearly, for some reason, every single night, Sherlock deleted him.</p>
            </blockquote>





	The Cost of Decreased Variance

**Author's Note:**

> Uh, I honestly didn't even proofread this one, so I hope it's okay. Um, the math came through a lot more in concept than it did in the actual writing, I think. But I guess it's enough to help me remember, which I s'pose is the important thing.
> 
> Also I want to note that on two of the previous drabbles, [Leverage](http://archiveofourown.org/works/545617) and [Influence](http://archiveofourown.org/works/549815), there have been some slight corrections to the math. Apparently my prof had accidentally swapped a couple of the terms when talking about them to us. XD

The _covratio_ is a function used in statistics in order to determine the effects of adding a particular point to a model. It is a function of the hat value (see [Influence](http://archiveofourown.org/works/549815)) as well as a measure of the error called the “studentized” error. This function is based on the idea that if you add a data point to your model that is an outlier, the variance of the estimators should increase—they will become less precise. The covratio is written as

 

COVRATIOi = 1/(1-hi) * 1/[(n-k-2+Ei*2)/n-k-1]k+1

Ordinarily we would expect that adding extra data to the regression will make the variance smaller (see [A Penalty for Profusion](http://archiveofourown.org/works/509493)), so if adding data makes the variance bigger (in other words, if covratio is close to 1, or bigger, that is the expected result; if it gets smaller, because not adding the point is better than adding the point, then we know that the point must be rather weird).

 

This is a means of measuring how influential (see [Leverage](http://archiveofourown.org/works/545617) and [Influence](http://archiveofourown.org/works/549815)) a point is in a regression—that is, how much its being there will affect the model. It doesn’t necessarily indicate that we need to get rid of the point/variable, because there may be a reason it’s an outlier that we are not taking into account (i.e. a different type of regression is needed, or a gap in data needs to be filled in), so that needs to be considered before simply removing the point from the regression in order to get smaller errors and variance.

 

  
***  
  
  
            There were a number of blog entries that John saved, but didn’t post.

            Really, there was no good reason for them to be in his blog at all; he could save them as text documents elsewhere, or maybe just type emails to himself, for all saving them in his blog drafts got him; god knew Sherlock would find them no matter where he put them.

            Then again, that was the point. And yes, maybe it was sentiment, a bit, that he chose his blog over any of the other perfectly viable options. John Watson had two journals, after all: one for his therapist, one for himself and Sherlock.

            In John’s blog for his therapist, he wrote sparse details about patients he treated, pleasant little metaphors about how being a locum doctor reminded him of the pleasanter bits of being in the army. (John had liked almost, _almost_ all of being in the army, but that wasn’t what his therapist wanted to hear, so John wrote about saving lives and not about how empty the nights before Sherlock had been, devoid of mad chases and gunshots and mystery.)

            In John’s blog for himself and Sherlock, he wrote about—well. Himself and Sherlock. Little notes for Sherlock to see when he inevitably gave in to the pull and logged into John’s open laptop each morning—reminders.

            _Pick up milk while you’re out_ , said the one John was currently typing, _and yes, I’m your flatmate, see drafts 1 through 325._

He started adding,

            _Please don’t delete me agai_

And then erased it and logged off the site and then off the computer. He knew Sherlock would guess the passwords because he had every day for almost the past year. If Sherlock were clever, he would write a sticky note to himself and put it on the refrigerator, but Sherlock was _too_ clever, so he never did, always reworked them. John kept it easy—nothing Sherlock couldn’t deduce about him from his possessions around the flat.

            Before he headed out to work ( _no case today, god, this evening will be interesting_ ) John peeked in to see Sherlock amidst his daily but very brief window of sleep. “Maddening git,” he muttered, and left.

 

 

 

            Sherlock scrolled backward through the entries.

            _Pick milk up while you’re out, and yes, I’m your flatmate, see drafts 1 through 325._

_I moved your toes to the next drawer over, don’t worry, it isn’t you losing your mind._

_Please, if you can help it, do not text Lestrade today. He and I had a bit of a chat while you were off interrogating that poor old woman about the shop she bought her buttons from and he’s taking today off and needs some time to himself._

_Sherlock, last night I came home a bit not sober and you threw a blanket over me before you went to bed. Thanks._

_Could you consider_

_Can we try not getting shot at today? Let’s try not getting shot at today. If you find a man dead of a heart attack upstairs, that’s me._

_I swear upon every eyeball you’ve stuffed into that bag in the fridge that if you don’t replace the contaminated jam by the time I get home, I will personally kill you._

_Why do you_

_In case you’ve forgotten, and yes, that was meant to be a bad joke, the jumpers upstairs are_ mine _. As in, not to be used in any variety of experiments. If you need wool or whatever, go buy your own bloody jumpers._

_Please_

_For such a brilliant bloke, you really have no idea what people are trying to say to you sometimes. Did you know that? Could you listen to me, for once, Sherlock?_

_Sherlock, you’re fantastic, but I really hate the fact that your brain is like a hard drive._

_If you read this, will you please reorganize the bookshelf by author? Not all of us have memorized which books have how many pages._

Sherlock opened up the first draft ever saved, and ran his hands through his hair. It was from January. He scanned over it, bits catching his eye, catching his throat.

            _…I definitely think he might be mad but he was also strangely likeable. He was charming. It really was all just a bit strange. So tomorrow, we're off to look at a flat. Me and the madman. Me and Sherlock Holmes. 1_

January.

            Nearly a year ago.

            Sherlock had awoken wondering whose things were in 221B; clearly, they were John Watson’s.

            Clearly, John Watson was his flatmate.

            And clearly, for some reason, every single night, Sherlock deleted him.

 

 

 

            “I’m back,” John leaned in through the door, “and also not a client, thief, or murderer, incidentally.”

            “John,” Sherlock said, and under his gaze John felt the daily scan begin. Sherlock always re-deduced everything; but since he had started reading John’s blog drafts, he had stopped doing it all out loud, realizing John had heard it a hundred times before. They hadn’t always been the same, of course—but John was, sometimes, glad not to be frozen in the doorway, listening to Sherlock deduce.

            “So, what do you remember?” It was a rather stupid question, and Sherlock generally remembered almost anything besides incidents directly involving John. He remembered _someone_ doing _something_ to help him at a crime scene; he remembered being held up waiting for _someone_ before he could dash off. He remembered the old cases; he remembered Jim Moriarty. He remembered that somehow, at the pool, Jim had done something that had disconcerted him greatly. John assumed it must not have been _such_ a huge setback, as Sherlock always seemed perfectly capable of solving anything that was thrown at him. The thought was disheartening.

            “I wrapped up a case yesterday,” Sherlock said, “involving, of all things, a faulty doorbell.”

            “Right, good.”

            “You needn’t quiz me.” But John could see from the curious gleam in Sherlock’s eye that he was working it right now, whether this was what they always did, every day.

            They did.

            John didn’t _have_ to stick around. By all rights, he should have found the idea of living with someone who decided John wasn’t worth remembering on a daily basis abhorrent. But it was interesting—different. And Sherlock was brilliant. It was terrible, and terrifying, and saddening. John would start each morning wondering whether he ought to move out; certainly, if he timed it right, Sherlock would never notice the difference. He could go on with his life, now that he had work, and live on his own, or maybe with someone who’d wake up and know his name in the morning. In the morning, John questioned why it was he didn’t just move out, why Sherlock was worth the hassle. By the end of the night, though, after the deductions or the chasing or the takeaway Thai or the giggling madly over things that one generally oughtn’t giggle at, by the time Sherlock had relearned John, by the time his eyes sparkled when he looked at John with the fondness of finding a friend, the newness of some sort of swooping infatuation (and that, John had decided, by now, was what it was), the electric energy of two souls bumping up against each other in camaraderie, by the time Sherlock finally realized _why_ John was his flatmate and _why_ John had written those journal entries, by _that_ time, John was in love.

             “I read your journal drafts,” Sherlock finally said, and John realized that they had locked eyes minutes ago and stared at one another, unspeaking. “I got the milk.”

            “Thank you.”

            These moments, these just-meeting moments, particularly on days when John wasn’t swept out the door for a case by a Sherlock that had concluded that John generally did this sort of thing, were always the strangest. John knew Sherlock better than anyone: Sherlock knew of John only what he had had time and resources to deduce throughout the day. Every day, John watched Sherlock attempt to wrap his mind around the idea that someone would willingly stay with him through this. “I don’t know why I do it, John,” Sherlock said. “Why I delete you.” This was what Sherlock always said, when they were just meeting again for the first time. He always said so, and John believed it; but by the end of the night, Sherlock had always found the reason.

 

 

 

            That night, during all the time that not having a case to work on bought them, John told Sherlock what John himself had done on their last case, about Sherlock trying to convince John to lock him into a car boot, about John nearly spilling pasta down his shirt when the murderer had actually (unknowingly) approached them in the restaurant.

            “That makes a great deal more sense,” Sherlock said. And then, as usual, as often in these nights when they would sit in the flat and discuss things Sherlock had deleted, Sherlock would lean in and ask quietly, “What about the pool?”

            “Moriarty strapped bombs to me,” John would explain, and then he would tell the story and Sherlock’s eyes, at some point, would glaze over slightly. Every morning, Sherlock learned about John. Every night, every mad errand that John followed him on or every person who threatened to kill Sherlock who John shot, or every deep and revealing discussion, the fond flickering of the flat’s lights in John’s eyes as he remembered hours and hours that Sherlock didn’t, the quietest or most amusing or most warming parts of the tales that Sherlock could never fill in—every evening, his mind refilling, Sherlock would fall in love with John.

            But every night, he deleted him.

            It was too much—too much to process. To different, too strange. Maybe if he only hadn’t done it the first few times, he wouldn’t have so much to catch up on. As it was, John threw everything off—sometimes the game wasn’t about the game, turned into being about John instead, and the thought was frightening. John said Sherlock had done some truly stupid things in the interest of saving John. Sherlock was horrified at the idea, at someone having so much control, this _one person_ dictating so much about his life and his decisions. Every clear sign pointed two ways: one, to the thrill of the game, and another, to the safety and caring and keeping of John Watson. The lack of certainty was maddening.

            So Sherlock deleted him.

            But right now, Sherlock was sitting on the sofa with John while John typed some shite about an old man with a heart condition onto his blog. John didn’t write about Sherlock; John didn’t want to answer questions about Sherlock. He’d told Greg about it, Sherlock was sure; Mycroft had to know. (Mycroft doubtless treated John horribly as a result: knowing that Sherlock deleted John every night, Mycroft, too, would be disconcerted by the sway John held, by the extent to which Sherlock’s action were affected by him.) It was a sheer stroke of luck that John hadn’t posted his first entry before meeting Sherlock again the next day and finding himself deleted. Sherlock must have explained the process then—the deleting.

            “I’d write about our case, if I could,” John said.

            Right now, Sherlock was sitting on the sofa with John; right now, Sherlock knew more about John than he’d known any other day before. Right now, Sherlock’s chest was swelling with the need to grab onto this man and find out what about him allowed him to put up with this; why, after almost a year, John Watson was still here.

            Right now, Sherlock was sitting on the sofa with John, in love with John.

            “I want to remember you,” Sherlock said.

            “Then don’t delete me.” John said it with the attention of a person who had had this discussion before. He looked up at Sherlock. “You’ll say you won’t, now, by the way, but you will.” His voice cracked slightly as he spoke. “It’s fine, though.”

            “It’s not.”

            John shrugged and resumed his typing, his feigned nonchalance cracking as his voice had as he swallowed down lumps in his throat, massaged away knots in his leg.

            “I have an idea.”

            Immediately, John’s eyes darted back up.

            “Don’t let me rationalize it away, John. Don’t let me think it’s the most logical choice. When I go to bed tonight, keep me from deleting you.”

            John swallowed, licked his lips. “And how would I do that?”

            “Come to bed with me.”

 

 

 

            John climbed tentatively into the covers. However strange this was for him, it must have been stranger for Sherlock, inviting a man he’d, for all intents and purposes, just met, into his bed. Not for anything—well. Not for anything like _that_ , of course.

            “Do you really think this will work?” John asked. Sherlock had never made such a request before. John fought his hopes back down; probably Sherlock would wake up confused and pissed off—and rightly so, with a stranger in his bed.

            “Take my hand,” Sherlock said, and John wove his fingers through it. He stayed awake for as long as he could, watching Sherlock, waiting for the inevitable moment when he’d close his eyes for daily deletion of any unnecessary _stuff_ and then open them again to look at John with a total lack of recognition.

            Instead, Sherlock stared back at John with cautious eyes until John drifted off.

 

 

 

            When John opened his eyes the next morning, the bed was empty.

            Well, it had been worth a shot. Sherlock had woken up, been confused, maybe felt awkward, maybe wondered what he and John had _done_ last night, and then left to avoid thinking about it at all. John rolled out of bed and pulled his trousers back on over his pants.

            The sitting room and kitchen were empty; Sherlock’s coat was gone.

            John sat down in front of his laptop, cupping his face in his hands for a few moments and taking slow, deep breaths. What would he write today? _You almost remembered me last night, let’s try again_ or maybe _You almost remembered me last night, and I can’t handle hoping like that again,_ or maybe _I can’t handle this, Sherlock, I’m leaving_ or maybe just _Thanks for getting the milk yesterday, I haven’t checked the refrigerator yet though so maybe you’ve got something growing in there again and I should take my thanks back._

Sherlock was already gone, but if he came back before John arrived back at the flat, he’d probably want some sort of an explanation of who John was and why he’d woken up in bed with him. John would just have to…write whatever came to mind. He logged into his blog and pulled up his drafts. There was an entry he hadn’t written. John opened it.

_Your heart rate varied from 54 to 76 beats per minute over the course of 11pm to 3am and 6am to 7am. Yes, I slept. Expect to take further measurements this evening. –SH_

**Author's Note:**

> 1: Portion of text for John's first saved draft taken directly from [The Personal Blog of Dr. John H. Watson](http://johnwatsonblog.co.uk/).


End file.
